Found problems: 56
Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant.
[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]
Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant.
[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]
Let $a,b,c,d$ be integers such that the number $a-b+c-d$ is odd and it divides the number $a^2-b^2+c^2-d^2$. Show that, for every positive integer $n$, $a-b+c-d$ divides $a^n-b^n+c^n-d^n$.
$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions:
[b]I.[/b] $f(1) = 2$
[b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$
Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.
Let $f$ be a function with the following properties:
1) $f(n)$ is defined for every positive integer $n$;
2) $f(n)$ is an integer;
3) $f(2)=2$;
4) $f(mn)=f(m)f(n)$ for all $m$ and $n$;
5) $f(m)>f(n)$ whenever $m>n$.
Prove that $f(n)=n$.
Show that $\cos 1^{\circ}$ is irrational.
The function $f : \mathbb N \to \mathbb R$ satisfies $f(1) = 1, f(2) = 2$ and \[f (n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ).\] Show that $0 \leq f(n+1) - f(n) \leq 1$. Find all $n$ for which $f(n) = 1025$.
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]
Consider a positive integer $a > 1$. If $a$ is not a perfect square then at the next move we add $3$ to it and if it is a perfect square we take the square root of it. Define the trajectory of a number $a$ as the set obtained by performing this operation on $a$. For example the cardinality of $3$ is $\{3, 6, 9\}$.
Find all $n$ such that the cardinality of $n$ is finite.
The following part problems may attract partial credit.
$\textbf{(a)}$Show that the cardinality of the trajectory of a number cannot be $1$ or $2$.
$\textbf{(b)}$Show that $\{3, 6, 9\}$ is the only trajectory with cardinality $3$.
$\textbf{(c)}$ Show that there for all $k \geq 3$, there exists a number such that the cardinality
of its trajectory is $k$.
$\textbf{(d)}$ Give an example of a number with cardinality of trajectory as infinity.
Let $\alpha $ be an angle such that $\cos \alpha = \frac pq$, where $p$ and $q$ are two integers. Prove that the number $q^n \cos n \alpha$ is an integer.
Given a positive integer number $k$, define the function $f$ on the set of all positive integer numbers to itself by
\[f(n)=\begin{cases}1, &\text{if }n\le k+1\\ f(f(n-1))+f(n-f(n-1)), &\text{if }n>k+1\end{cases}\]
Show that the preimage of every positive integer number under $f$ is a finite non-empty set of consecutive positive integers.
Let $f$ be a function with the following properties:
1) $f(n)$ is defined for every positive integer $n$;
2) $f(n)$ is an integer;
3) $f(2)=2$;
4) $f(mn)=f(m)f(n)$ for all $m$ and $n$;
5) $f(m)>f(n)$ whenever $m>n$.
Prove that $f(n)=n$.
Let $ x,y$ be real numbers such that the numbers $ x\plus{}y, x^2\plus{}y^2, x^3\plus{}y^3$ and $ x^4\plus{}y^4$ are integers. Prove that for all positive integers $ n$, the number $ x^n \plus{} y^n$ is an integer.
Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?
Prove that every positive integer can be represented in the form $$3^{m_1}\cdot 2^{n_1}+3^{m_2}\cdot 2^{n_2} + \dots + 3^{m_k}\cdot 2^{n_k}$$
where $m_1 > m_2 > \dots > m_k \geq 0$ and $0 \leq n_1 < n_2 < \dots < n_k$ are integers.
Consider a regular $n$-gon with $n$ odd. Given two adjacent vertices $A_{1}$ and $A_{2},$ define the sequence $(A_{k})$ of vertices of the $n$-gon as follows: For $k\ge 3,\, A_{k}$ is the vertex lying on the perpendicular bisector of $A_{k-2}A_{k-1}.$ Find all $n$ for which each vertex of the $n$-gon occurs in this sequence.
Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions:
(1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$;
(2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$;
(2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$.
Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.
The numbers $1, 2, \ldots, 2012$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number $N$ remains. Find the remainder when the maximum possible value of $N$ is divided by 1000.
[i]Victor Wang.[/i]
Let $ \{x_n\}_{n\geq 1}$ be a sequences, given by $ x_1 \equal{} 1$, $ x_2 \equal{} 2$ and
\[ x_{n \plus{} 2} \equal{} \frac { x_{n \plus{} 1}^2 \plus{} 3 }{x_n} .
\]
Prove that $ x_{2008}$ is the sum of two perfect squares.
Prove that for every real number $M$ there exists an infinite arithmetical progression of positive integers such that [list] [*] the common difference is not divisible by $10$, [*] the sum of digits of each term exceeds $M$. [/list]
Consider a convex polygon having $n$ vertices, $n\geq 4$. We arbitrarily decompose the polygon into triangles having all the vertices among the vertices of the polygon, such that no two of the triangles have interior points in common. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the triangle is also a side of the polygon and in white those triangles that have no sides that are sides of the polygon.
Prove that there are two more black triangles that white ones.
There is a $2012\times 2012$ grid with rows numbered $1,2,\dots 2012$ and columns numbered $1,2,\dots, 2012$, and we place some rectangular napkins on it such that the sides of the napkins all lie on grid lines. Each napkin has a positive integer thickness. (in micrometers!)
(a) Show that there exist $2012^2$ unique integers $a_{i,j}$ where $i,j \in [1,2012]$ such that for all $x,y\in [1,2012]$, the sum \[ \sum _{i=1}^{x} \sum_{j=1}^{y} a_{i,j} \] is equal to the sum of the thicknesses of all the napkins that cover the grid square in row $x$ and column $y$.
(b) Show that if we use at most $500,000$ napkins, at least half of the $a_{i,j}$ will be $0$.
[i]Proposed by Ray Li[/i]
A positive integer is written on a blackboard. Players $A$ and $B$ play the following game: in each move one has to choose a proper divisor $m$ of the number $n$ written on the blackboard ($1<m<n$) and replaces $n$ with $n-m$. Player $A$ makes the first move, then players move alternately. The player who can't make a move loses the game. For which starting numbers is there a winning strategy for player $B$?
Let $x_1,\ldots ,x_n$ be positive real numbers. Show that there exist $a_1,\ldots ,a_n\in\{-1,1\}$ such that:
\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_n)^2\]